Solve frac{1+sqrt{5}}{2+5/sqrt{5}}

$$\frac{ 1+ \sqrt{ 5 } }{ 2+ \frac{ 5 }{ \sqrt{ 5 } } }$$

Evaluate

$$3-\sqrt{5}\approx 0.763932023$$

Solution Steps

Rationalize the denominator of \frac{5}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{1+\sqrt{5}}{2+\frac{5\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{1+\sqrt{5}}{2+\frac{5\sqrt{5}}{5}}$$

Cancel out 5 and 5.

$$\frac{1+\sqrt{5}}{2+\sqrt{5}}$$

Rationalize the denominator of \frac{1+\sqrt{5}}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.

$$\frac{\left(1+\sqrt{5}\right)\left(2-\sqrt{5}\right)}{\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)}$$

Consider \left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(1+\sqrt{5}\right)\left(2-\sqrt{5}\right)}{2^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 2. Square \sqrt{5}.

$$\frac{\left(1+\sqrt{5}\right)\left(2-\sqrt{5}\right)}{4-5}$$

Subtract 5 from 4 to get -1.

$$\frac{\left(1+\sqrt{5}\right)\left(2-\sqrt{5}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\left(1+\sqrt{5}\right)\left(2-\sqrt{5}\right)$$

Apply the distributive property by multiplying each term of 1+\sqrt{5} by each term of 2-\sqrt{5}.

$$-\left(2-\sqrt{5}+2\sqrt{5}-\left(\sqrt{5}\right)^{2}\right)$$

Combine -\sqrt{5} and 2\sqrt{5} to get \sqrt{5}.

$$-\left(2+\sqrt{5}-\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$-\left(2+\sqrt{5}-5\right)$$

Subtract 5 from 2 to get -3.

$$-\left(-3+\sqrt{5}\right)$$

To find the opposite of -3+\sqrt{5}, find the opposite of each term.

$$-\left(-3\right)-\sqrt{5}$$

The opposite of -3 is 3.

$$3-\sqrt{5}$$